**3. Dimensionless Formulations Model**

Similarity transformations that convert the governing PDEs into ODEs, and the BVP formulae (3)–(7) are modified. Familiarizing stream function *ψ* in the equation [28]

$$G\_1 = \frac{\partial \psi}{\partial y'},\ G\_2 = -\frac{\partial \psi}{\partial x}.\tag{9}$$

The specified similarity quantities are ([28])

$$\gamma^\*(\mathbf{x}, y) = \sqrt{\frac{m}{\nu\_f}} y, \quad \psi(\mathbf{x}, y) = \sqrt{\nu\_f m} \mathbf{x} f(\gamma^\*), \quad \theta(\gamma^\*) = \frac{\mathbb{Y} - \mathbb{Y}\_{\infty}}{\mathbb{Y}\_w - \mathbb{Y}\_{\infty}}. \tag{10}$$

into Equations (3)–(7). We ge<sup>t</sup>

$$\tau^\* f^{\prime\prime} \left( 1 - \varsigma^\* f^{\prime 2} \right) + \phi \varsigma\_2 \left[ f f^{\prime} - f^{\prime 2} \right] - \frac{1}{\phi\_{\hat{\mathcal{Y}}\_1}} F\_{\pi} f^{\prime} = 0,\tag{11}$$

$$\theta'' \left( 1 + \upsilon^\* \theta + \frac{1}{\Phi\_{\mathbf{Y}\_4}} P\_r N\_\pi \right) + \upsilon^\* \theta'^2 + P\_r \frac{\phi\_{\mathbf{Y}\_3}}{\Phi\_{\mathbf{Y}\_4}} \left[ f \theta' - f' \theta \right] = 0. \tag{12}$$

with

$$\begin{array}{c} f(0) = \mathcal{S}, \; f'(0) = 1 + \Lambda\_{\pi} f''(0), \theta'(0) = -B\_{\pi} (1 - \theta(0)) \\\ f'(\gamma^\*) \to 0, \; \theta(\gamma^\*) \to 0, \; \text{as } \gamma^\* \to \infty \end{array} \tag{13}$$

where *φ*Υ¨ *i* is 1 ≤ *i* ≤ 4 in formulae (11)–(12) signify the subsequent thermophysical structures for P-ENF [29].

$$\begin{split} \boldsymbol{\Phi}\_{\bar{\mathbf{Y}}\_{1}} &= (1 - \boldsymbol{\phi})^{2.5}, \quad \boldsymbol{\Phi}\_{\bar{\mathbf{Y}}\_{2}} = \left(1 - \boldsymbol{\phi} + \boldsymbol{\phi}\frac{\boldsymbol{\rho}\_{\mathbf{z}}}{\boldsymbol{\rho}\_{f}}\right), \quad \boldsymbol{\Phi}\_{\bar{\mathbf{Y}}\_{3}} = \left(1 - \boldsymbol{\phi} + \boldsymbol{\phi}\frac{(\boldsymbol{\rho}\boldsymbol{C}\_{\mathcal{P}})\_{\mathbf{z}}}{(\boldsymbol{\rho}\boldsymbol{C}\_{\mathcal{P}})\_{f}}\right) \\ \boldsymbol{\Phi}\_{\bar{\mathbf{Y}}\_{4}} &= \left(\frac{\left(k\_{s} + 2k\_{f}\right) - 2\boldsymbol{\phi}\left(k\_{f} - k\_{s}\right)}{\left(k\_{s} + 2k\_{f}\right) + \boldsymbol{\phi}\left(k\_{f} - k\_{s}\right)}\right). \end{split} \tag{14}$$

Equation (2) is clearly shown to be valid. Table 3 shows the needed derivatives.

**Table 3.** Entrenched Control Constraints.


Other parameters like skin friction (*Cf*), Nusselt number (*Nux*) and entropy generation ( *NG*) can be expressed as [31,32]:

$$\begin{aligned} \mathbb{C}\_{f}\mathrm{Re}\_{\boldsymbol{x}}^{\frac{1}{2}} &= \tau^\* f''(0) - \frac{1}{3} \tau^\* \xi^\* \left( f''(0) \right)^3, \\ \mathrm{Nu}\_{\boldsymbol{x}} \mathrm{Re}\_{\boldsymbol{x}}^{-\frac{1}{2}} &= -\frac{k\_{\boldsymbol{n}}}{k\_{f}} (1 + N\_{\pi}) \theta'(0), \\ \mathrm{N}\_{\boldsymbol{G}} &= \mathcal{R}\_{\pi} \left[ \phi\_{\hat{\mathcal{Y}}\_{4}} (1 + N\_{\pi}) \theta'^2 + \frac{1}{\Phi\_{\hat{\mathcal{Y}}\_{4}}} \frac{\mathcal{B}\_{\mathbb{T}}}{\Pi} \left( f''^2 + P\_{\tilde{\xi}} f'^2 \right) \right]. \end{aligned} \tag{15}$$
